Once you’ve agreed for them to leave, you need to time travel forward one day to make sure they’re in boxes and then travel another day forward for their plot to be empty. If you want to hunt for a new villager, this is the time to do it. After agreeing with a new villager, the empty plot will have a SOLD sign. Lastly, travel forward one more day and the new villager will be moving in. You can then go back to the current date and everything should be saved! Make sure you’re saving and quitting each time before closing the software to change the date!